Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions

TL;DR

This paper develops new combinatorial formulas, called Murnaghan--Nakayama rules, for multiplying power sum symmetric functions with symplectic, orthogonal, and orthosymplectic Schur functions, expanding understanding of their representation theory.

Contribution

It introduces explicit Murnaghan--Nakayama rules for symplectic, orthogonal, and orthosymplectic Schur functions, including combinatorial and algebraic descriptions of the product formulas.

Findings

Derived explicit formulas for symplectic, orthogonal, and orthosymplectic Schur functions.

Formulas include classical border strip addition, removal, and a new third term.

Rules are described both algebraically and combinatorially.

Abstract

We establish new Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions. The classical Murnaghan--Nakayama rule expresses the product of a power sum symmetric function with a Schur function as a linear combination of Schur functions. Symplectic and orthogonal Schur functions correspond to characters of irreducible representations of symplectic and orthogonal groups. Orthosymplectic Schur functions arise as characters of orthosymplectic Lie superalgebras and are hybrids of symplectic and ordinary Schur functions. We derive explicit formulas for the product of the relevant power-sum function with each of these functions, which can partly be described combinatorially using border strip manipulations. Our Murnaghan--Nakayama rules each include three distinct terms: a classical term corresponding to the addition of border strips to the relevant Young diagram, a term involving the removal of border strips, and a third term, which we describe both algebraically and combinatorially.